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04:58
A Rapid Method for Modeling a Variable Cycle Engine
Published on: August 13, 2019
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A Gamma Ornstein-Uhlenbeck model driven by a Hawkes process.
Guillaume Bernis1, Riccardo Brignone2, Simone Scotti3
1Natixis Assurances, Paris, France.
Mathematics and Financial Economics
|April 16, 2024
Summary
We introduce a new financial model extending the -OU Barndorff-Nielsen and Shephard model to include jump clustering using Hawkes processes. This enhanced model better captures market dynamics, particularly for VIX options, showing an upward implied volatility slope.
Area of Science:
- Quantitative Finance
- Stochastic Processes
- Financial Modeling
Background:
- The -OU Barndorff-Nielsen and Shephard model is a cornerstone in financial econometrics but lacks the ability to capture jump clustering.
- Jump clustering phenomena significantly impact asset price dynamics and option pricing, necessitating more sophisticated models.
Purpose of the Study:
- To extend the -OU Barndorff-Nielsen and Shephard model by incorporating jump clustering via Hawkes processes.
- To analyze the mathematical properties and financial implications of this new -OU-Hawkes model.
- To demonstrate the model's improved flexibility and calibration capabilities.
Main Methods:
- Extension of the -OU model with a Hawkes process for jump clustering.
- Application of continuous-state branching processes with immigration for SDE solutions.
- Development of a self-exciting Esscher-type measure change for risk-neutral dynamics.
- Exploitation of affine properties for analytical solutions (Laplace transform, quadratic variation).
- Calibration using market vanilla option prices and characteristic function inversion.
Main Results:
- Existence and uniqueness of strong solutions for the asset price SDE.
- Explicit forms for key financial quantities like log-return Laplace transform and variance distribution.
- Demonstrated increased flexibility compared to the standard -OU model with an equivalent parameter count.
- Successful calibration to market data and development of an exact simulation scheme.
- Observed upward-shaped implied volatility for VIX options, contrasting with the -OU model's downward slope.
Conclusions:
- The proposed -OU-Hawkes model effectively captures jump clustering, offering enhanced flexibility in financial modeling.
- The model provides a more realistic representation of market behavior, particularly evident in the implied volatility of VIX options.
- This framework offers significant advancements for pricing and risk management in complex financial markets.
Keywords:
Exponential affine processesHawkes processesImplied volatility for VIX optionsJump clustersLeverage effectStochastic volatilityVIXMore Related Videos
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